Simple Hironaka resolution in characteristic zero

By J. Włodarczyk. 2004

80 citations recorded by the source corpus.

Building upon work of Villamayor and Bierstone-Milman we give a proof of the canonical Hironaka principalization and desingularization. The idea of "homogenized ideals" introduced in the paper gives a priori the canonicity of algorithm and radically simplifies the proof.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1746ff3f154a0987e12707494a50526d7710038b

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Building blocks of étale endomorphisms of complex projective manifolds (2009, methodology): “to the fiber product X ×Y Y � . The recent methods by Bierstone–Milman and by Villamayor using the notion of invariant have the functoriality (cf. [5, 14, 15, 39, 55 ]).”
Singularities in positive characteristic, stratification and simplification of the singular locus (2008, background): “f the choice of the hypersurface of maximal contact. This result is the main outcome of the so called Hironaka trick; an alternative and enlightening proof of this result is due to J. Wlodarczyk (see [31]). In this work hypersurfaces of maximal contact are replaced by suitable projections, and we be”
Rees algebras and resolution of singularities (2007, background): “o that fn has order n at OV,x. Note that ∆α(fn)W ∈G(G) for |α|= n −1, which can be chosen so that ∆α(f n) has order one, deﬁning a smooth hypersurface Z locally at x (3.3). Theorem 6.8. (W lodarczyk, [20]) Let x ∈V be a simple point of G, and assume that locally at x, there are two hypersurfaces Z1,”
Resolution of singularities of arithmetical threefolds (2014, background): “Hironaka’s celebrated Theorem [46] proved fifty years ago, some new results have bettered our understanding of the problem in equal characteristic zero [13][81][83].”
Resolution of singularities for 3-folds in positive characteristic (2006, background): “racteristic are [Co1], [Co2], [Hau1], [Jo], [Ku], [P], [S], [T]. Several simpliﬁcations of Hironaka’s proof of characteristic zero resolution have appeared, including [BrM], [BP], [BEV], [Hau2], [V], [W]. We prove the following theorem. Theorem 1.1. Suppose that V is a projective variety of dimensio”
On the problem of resolution of singularities in positive characteristic (Or: A proof we are still waiting for) (2009, background): “This program appears in different disguises in many places, see, e.g., [Hi5, Vi1, BM, EV1, EH, Wl , Ko].”